FLAT $\phi $ CURVATURE FLOW OF CONVEX SETS

Abstract

For an arbitrary initial compact and convex subset $K_{0}$ of $\mathbb{R}^{n},$ and for an arbitrary norm $\phi$ on $\mathbb{R}^{n}$, we construct a flat $\phi$ curvature flow $K\left(t\right)$ such that $K\left(t\right)$ is compact and convex throughout the evolution. Previously and using similar methods, R. McCann had shown that flat $\phi$ curvature flow in the plane preserves convex, balanced sets. More recently, G. Bellettini, V. Caselles, A. Chambolle, and M. Novaga showed that flat $\phi$ curvature flow in $\mathbb{R}^{n}$ preserves compact, convex sets. We also establish a new Hölder continuity estimate for the flow. Flat $\phi$ curvature flows, introduced by F. Almgren, J. Taylor, and L. Wang, model motion by $\phi$-weighted mean curvature. Under certain regularity assumptions, they coincide with smooth $\phi$-weighted mean curvature flows given by partial differential equations as long as the smooth flows exist.